von Neuman algebras of strongly connected higher-rank graphs

Abstract

We investigate the factor types of the extremal KMS states for the preferred dynamics on the Toeplitz algebra and the Cuntz--Krieger algebra of a strongly connected finite $k$-graph. For inverse temperatures above 1, all of the extremal KMS states are of type I${}_{\infty }$. At inverse temperature 1, there is a dichotomy: if the $k$-graph is a simple $k$-dimensional cycle, we obtain a finite type I factor; otherwise we obtain a type III factor, whose Connes invariant we compute in terms of the spectral radii of the coordinate matrices and the degrees of cycles in the graph.

16 pages; 1 picture prepared using TikZ. Version 2: this version to appear in Math. Ann

References in corpus (4)

• The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources
• On some fundamental results about higher-rank graphs and their C*-algebras
• Simplicity of the C*-algebras of skew product k-graphs
• Factoriality and type classification of \textsf{k}-graph von Neumann algebras

Cited by in corpus (8)

• Wavelets and spectral triples for higher-rank graphs
• Phase transitions on C*-algebras from actions of congruence monoids on rings of algebraic integers
• C*-dynamical invariants and Toeplitz algebras of graphs
• Separable representations of higher-rank graphs
• KMS states on the C*-algebras of Fell bundles over groupoids
• Equilibrium states on the Toeplitz algebras of small higher-rank graphs
• The classification of some generalised Bunce-Deddens algebras
• KMS states on generalised Bunce-Deddens algebras and their Toeplitz extensions